The Faber-Krahn position of convex bodies and Gaussian measure inequalities

Dmitry Faifman; Iosif Polterovich

The Faber-Krahn position of convex bodies and Gaussian measure inequalities

T0 review · 0 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Every convex body has a unique Faber–Krahn position, enforced by log-convexity of the first eigenvalue along linear flows.

desk verdict Strong spectral-geometry paper; the main theorem is new and the proof is credible, with the quasi-stationary estimate as the main thing to check. read the letter →

arxiv 2607.21539 v2 pith:ATWQWKSG submitted 2026-07-23 math.SP math.FAmath.MG

Dmitry Faifman , Iosif Polterovich This is my paper · ORCID

classification math.SPmath.FAmath.MG MSC 35P1552A4060J65

keywords firstDirichleteigenvaluelog-convexityFaber–KrahnpositionconvexbodiesBrownianmotionquasi-stationarydistributionGaussianmeasureinequalitiesaffine

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes a new convexity law for the first Dirichlet eigenvalue of a convex body: as the body is deformed by powers P^t of a positive-definite matrix, the value λ₁(P^t K) is log-convex in t. From this one-parameter rigidity the authors extract a global geometric fact: every convex body has a Faber–Krahn position—a volume-preserving linear image that minimizes λ₁—and that position is unique up to rotations, answering a question raised in 2011. The eigenvalue version needs no symmetry hypothesis because the proof replaces the Gaussian B-theorem with a quantitative analysis of Brownian motion conditioned to stay inside the deforming body; a uniform exponential-mixing estimate makes the obstruction to convexity vanish in the long-time limit. If correct, the result supplies a spectral counterpart to John's ellipsoid, recovers the classical triangle and simplex minimization theorems, and shows regular polygons are extremal within their own linear orbits. A second line uses a Gaussian Rogers–Shephard inequality to sharpen the classical intersection bound for centrally symmetric bodies to λ₁(K∩L)+λ₁(K+L) ≤ λ₁(K)+λ₁(L).

What carries the argument

The engine is the one-parameter family K_t = P^t K. For centrally symmetric K, log-convexity drops out of the Gaussian B-theorem applied to sampled Brownian paths. For general K, the proof rests on a uniform discrete quasi-stationary mixing estimate (Theorem 3.4): a Brownian path conditioned to stay in K_t at discrete times has conditional mean at time kh exponentially close to the quasi-stationary mean, uniformly over compact t-intervals. Together with an L² estimate for the second derivative of the sampled Gaussian log-volume and a discrete Kac formula, this makes the barycenter obstruction only o(T), so dividing by T restores convexity of λ₁.

What would settle it

Numerically compute, for a fixed non-symmetric K (say a right triangle) and P = diag(e^s, e^{-s}) with s ≠ 0, the second derivative of log λ₁(P^t K) at t = 0; log-convexity predicts it is nonnegative for every s and strictly positive when s ≠ 0, so one negative value refutes Theorem 1.1. Alternatively, simulate the discretely conditioned Brownian motion and check whether the conditional mean in Theorem 3.4 converges exponentially; observing only polynomial convergence would invalidate the proof.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: for any convex body K and positive-definite P, t ↦ λ₁(P^t K) is log-convex on R, and strictly so when P preserves volume and is not the identity. This says the first eigenvalue never prefers an interpolated linear deformation over the interpolation of its own values. The centrally symmetric case follows from the Gaussian B-theorem; general bodies are handled by sampling Brownian motion at discrete times and conditioning on survival, with a uniform exponential-mixing estimate that makes the barycenter obstruction sublinear in time and lets Kac's formula pass to the limit. From this they derive Theorem 2.4: existence and uniqueness (up to orthogonal transforma

Load-bearing premise

The proof that log-convexity holds for all convex bodies rests on the uniform discrete quasi-stationary mixing estimate: Brownian motion sampled at discrete times and conditioned to stay inside the deforming body must mix exponentially fast to its quasi-stationary law, uniformly on compact parameter intervals; if that rate were only polynomial, the barycenter term would not become sublinear and the convexity limit would fail.

Editorial extensions

If this is right

Every convex body admits a Faber–Krahn position, and the minimizer is unique up to orthogonal transformations; the extremal condition is the isotropic energy identity ∫_K ∇ψ⊗∇ψ = (λ₁(K)/n) I for the first eigenfunction.
Bodies whose symmetry group acts irreducibly—regular simplex, regular polygons, regular cross-polytopes, Platonic solids—are automatically in Faber–Krahn position, so they minimize λ₁ among all their volume-preserving linear images.
The regular simplex minimizes λ₁ among all n-simplices of fixed volume in every dimension, giving a new proof of the triangle and simplex cases of Pólya–Szegő; regular N-gons likewise minimize among their area-normalized linear images.
For centrally symmetric K, L, the improved Schmuckenschläger inequality λ₁(K∩L)+λ₁(K+L) ≤ λ₁(K)+λ₁(L) holds, and the same inequality holds for the first Dirichlet eigenvalue of the Ornstein–Uhlenbeck operator.
Log-convexity also holds for the inverse inradius and for the planar Cheeger constant; for centrally symmetric bodies, the heat trace of P^t K is log-concave in t.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the Faber–Krahn position is as canonical as John's position, it gives a GL-equivariant way to attach a Euclidean metric (a 'spectral ellipsoid') to a convex body; this could serve as a shape-normalization tool in computational geometry and image analysis.
The same quasi-stationary machinery might transfer to other spectral quantities, such as the first nonzero Neumann eigenvalue, if its conjectured log-concavity (Question 5.5) holds; the paper's open list explicitly invites this.
The p-Laplacian version is open; a positive answer would interpolate between the John position (p=∞, inradius) and the spectral position (p=2), suggesting a continuous family of affine positions.
The improved Schmuckenschläger inequality is derived from a Gaussian inequality with no Brownian conditioning; the same inequality may strengthen other survival-probability bounds for nonsymmetric bodies once centering conditions are understood.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

I read the Faifman–Polterovich paper. Short version: this is a serious paper and the reader's ACCEPT is right. The main result — log-convexity of λ1(P^t K) for arbitrary convex bodies, strict for non-orthogonal P in SL(n) — is genuinely new. The centrally symmetric case falls out of the Gaussian B-theorem, but the general case needs a real new ingredient: the quantitative quasi-stationary estimate for discretely killed Brownian motion, which controls the barycenter term in the L2 method. That step is the heart of the paper, and it is the one a referee should read carefully.

What the paper does well: the proof is structured and honest. The discrete Kac formula (Proposition 3.9) is stated cleanly; the L2 estimate follows the Cordero-Erausquin–Fradelizi–Maurey method and is re-derived correctly; the applications (regular simplex, regular polygons in their SL(2)-orbits, cross-polytopes) are immediate but nice consequences of the irreducible symmetry principle. The paper also marks its open questions sharply — the p-Laplacian case, the first Neumann eigenvalue — without overclaiming.

Where the soft spots are, in proportion: first, Theorem 3.4 is the load-bearing estimate. The stress-test note did a second pass on the step at §3.6 where it is used, and found no internal gap; I agree that the proof is coherent. But it is intricate, and the uniformity in t depends on compactness arguments that are plausible but not trivial. A referee should verify Lemmas 3.6–3.8 and the error control in (28) line by line. That is the one place where a hidden mistake would topple the main theorem.

Second, Theorem 1.3 — the improved Schmuckenschläger inequality — relies on the Gaussian conjugate Rogers–Shephard inequality of Milman–Nakamura–Tsuji, which is itself a 2026 preprint. The implication from (6) to (7) is a one-line Brownian-sampling argument, so the dependency is external but not deep. Still, the referee should note that a secondary result of the paper is conditional on an unverified preprint.

Third, the heat-trace log-concavity (Theorem 2.21) is only sketched — the regularization of the loop measure is mentioned, not carried out. That is minor, since it is not central to the paper's main contribution.

Citation pattern is fine: they build on Brascamp–Lieb, Cordero-Erausquin–Fradelizi–Maurey, Schmuckenschläger, and the relevant convex-geometry literature. No self-citation inflation.

Who should read it: anyone working on spectral geometry of convex bodies, affine positions of convex sets, or Gaussian measure inequalities. I would bring it to a reading group and I would cite the main theorem.

Recommendation: send it to serious peer review. The main theorem deserves the referee time. Expect heavy scrutiny of §3.3–3.6, but that is what peer review is for.

Referee Report

0 major / 4 minor

Summary. The paper proves that for any convex body K in R^n and any positive definite matrix P, the first Dirichlet eigenvalue f(t)=λ1(P^tK) is log-convex on R, and strictly log-convex when P∈SL(n) is non-orthogonal. From this it derives that every convex body admits a unique (up to orthogonal transformations) Faber–Krahn position: a volume-preserving linear image minimizing λ1 over the SL(n)-orbit. This answers a question of Schmuckenschläger. Further applications include affine-orbit minimization results for simplices, centrally symmetric polytopes with 2n vertices, and regular polygons; log-convexity of the inverse inradius and planar Cheeger constant; heat-trace log-concavity in the centrally symmetric case; convexity for the Ornstein–Uhlenbeck first eigenvalue; and an improved Schmuckenschläger-type inequality λ1(K∩L)+λ1(K+L)≤λ1(K)+λ1(L) for centrally symmetric convex bodies, using the Gaussian conjugate Rogers–Shephard inequality.

Significance. The main theorem is a new and striking spectral- geometric convexity property: log-convexity of λ1 under positive definite linear flows. It resolves the existence and uniqueness of the Faber–Krahn position, a natural spectral analogue of John's position, and gives clean proofs of several affine-orbit eigenvalue minimization results. The proof is substantial and mostly self-contained: it introduces a uniform discrete quasi-stationary mixing estimate (Theorem 3.4), a discrete Kac formula (Proposition 3.9), and a quantitative L2 estimate extending the Cordero–Erausquin–Fradelizi–Maurey method. I found the central derivation coherent and the key estimate internally consistent. The paper is not machine-checked, but the main argument is presented with explicit quantitative statements and locally uniform constants, which is a notable strength. The main advertised results are falsifiable and likely to inspire further work on affine positions and p-Laplacian analogues.

minor comments (4)

[§3.10 / Theorem 2.21] The proof of heat-trace log-concavity is only a sketch: the Brownian-loop discretization is stated without derivation, and the regularization step 'adding η∑|x_j|²' followed by η↓0 is not justified in detail. Since Theorem 2.21 is not used in the main Faber–Krahn argument, this does not affect the central claim, but the authors should either expand the proof or explicitly downgrade the statement to a remark/conjecture with a heuristic proof.
[§4.1 / Theorem 1.3] Theorem 1.3 is an advertised consequence, but its proof is a direct application of inequality (6) from the external preprint [32] (Milman–Nakamura–Tsuji), which is not proved in the manuscript. I recommend the authors state explicitly in the introduction that this part is conditional on [32], or include a proof of (6). This does not affect the correctness of the central Faber–Krahn result.
[Corollary 2.3] There is a typo in the final reduction step: the definition of P′ should read P′ = ((1−t0)In+t0P)^{-1}((1−t1)In+t1P). The matrix AM-GM step leading to c_{t,s}P^s would also benefit from a brief explanation, since it is central to the proof of the displayed inequality.
[§3.3.5 / Theorem 3.4] The absorption of the boundary cases kh≤TK or (m−k)h≤TK into the constant CK is correct, but a one-sentence justification would improve readability: in these cases the exponential sum is bounded below by e^{-c_K T_K}, so CK can be enlarged to dominate the O(1) L1 error. The current wording 'can be accommodated by adjusting CK' is a little terse for such a technical step.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: central derivation self-contained; self-citation only background.

full rationale

The central derivation is not circular. Theorem 1.1 is proved from the external Gaussian B-theorem in the centrally symmetric case and, for general convex bodies, from a quantitative Brownian discretization culminating in the uniform quasi-stationary estimate (Theorem 3.4). The latter is proved in the paper from spectral-gap and Perron-eigenfunction estimates (Lemmas 3.5–3.8), with the discrete-to-continuous convergence established by min-max arguments in Proposition 3.5. The L2 estimate (Lemmas 3.10–3.11) is a Brascamp-Lieb-type inequality whose proof is included. Strict log-convexity follows from properness (Proposition 3.12) and real analyticity (Lemma 3.13), not from the desired conclusion. Uniqueness of the Faber-Krahn position is a consequence of strict log-convexity, not an input. No fitted parameter is renamed as a prediction, and no load-bearing uniqueness theorem is imported from the authors' prior work. The only self-citation is [30], a book co-authored by one of the present authors, used to point to a stated conjecture as background and not load-bearing. The main risk is the complexity of Theorem 3.4, which is a verifiability concern rather than circularity.

Assumptions & free parameters 0 free parameters · 10 assumptions · 0 invented entities

No parameters are fitted. No new physical entities are postulated; the Faber–Krahn position and ellipsoid are derived definitions. The central theorem rests on external results: the B-theorem, Kac's formula, Krein-Rutman, Kato perturbation theory, and for the applications the Gaussian conjugate Rogers–Shephard inequality (a very recent preprint).

assumptions (10)

standard math Strong B-theorem: for a centrally symmetric convex body A in Gaussian space and any symmetric S, t↦γ(e^{tS}A) is log-concave.
Used in Lemma 3.2 and Theorem 3.3 to prove the centrally symmetric case of Theorem 1.1, and in Theorem 2.21; cited to [9] and [20].
standard math Kac's formula linking first exit time survival probability to λ1 (equation (1)).
Bridges Brownian survival and the first Dirichlet eigenvalue throughout Sections 2 and 3.
domain assumption Gaussian conjugate Rogers–Shephard inequality (6) of Milman–Nakamura–Tsuji [32]: for centrally symmetric convex sets A,B and centered Gaussian γ, γ(A)γ(B)≤γ(A∩B)γ(A+B).
Black-box input for Theorem 1.3 and Theorem 4.1; a very recent preprint not proved in this paper.
standard math Payne–Stakgold lower bound λ1(K)≥π^2/(4 r(K)^2) for convex K.
Used in Proposition 3.12 to prove properness of the SL(n)-orbit, needed for existence and strict log-convexity.
standard math Brascamp–Lieb Brunn–Minkowski inequality for λ1 (eq. (3)): λ1^{-1/2} is concave under Minkowski addition.
Used in Corollary 2.3, Theorem 4.3 and Section 5; cited [10].
standard math John's theorem on the unique maximal volume ellipsoid.
Used in Proposition 2.16 for uniqueness of the inradius-maximizing position.
standard math Kawohl–Lachand-Robert planar Cheeger set characterization |Ω⊖r|=πr^2.
Used in Proposition 2.19 for log-convexity of the planar Cheeger constant.
standard math Kato analytic perturbation theory for simple eigenvalues of holomorphic families of forms.
Used in Lemma 3.13 to show t↦λ1(P^tK) is real analytic, needed for strict log-convexity.
standard math Krein–Rutman theorem: a positivity improving compact operator has a positive principal eigenfunction.
Used in §3.3.3 to justify the quasi-stationary distribution φ_h.
standard math Andrews–Clutterbuck log-concavity/fundamental gap estimate giving unique maximum of the first eigenfunction.
Used in Lemma 2.10 to define the Faber–Krahn ellipsoid.

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Pith. "Pith review of The Faber-Krahn position of convex bodies and Gaussian measure inequalities." pith.science (2026). https://pith.science/paper/ATWQWKSG

@misc{pith2026260721539,
  author       = {Pith},
  title        = {Pith review of: The Faber-Krahn position of convex bodies and Gaussian measure inequalities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ATWQWKSG}},
  note         = {Machine review of arXiv:2607.21539}
}

read the original abstract

We say that a convex body is in Faber-Krahn position if it minimizes the first Dirichlet eigenvalue within its volume-preserving linear orbit. We prove that this position is unique up to orthogonal transformations, answering a question of Schmuckenschlaeger from 2011. This is a corollary of a new log-convexity property of the first eigenvalue under positive definite linear deformations. While the centrally symmetric case follows from the Gaussian B-theorem, the extension to arbitrary convex bodies requires a quantitative analysis of conditioned Brownian motion. As consequences, we obtain a new proof of the Polya-Szego theorem for triangles and its analogue for simplices, and show that regular polygons minimize the first eigenvalue within their linear orbits of fixed volume. We also prove related convexity results for the first eigenvalue of the Ornstein-Uhlenbeck operator, the inverse inradius and the planar Cheeger constant. In a different direction, we show using similar ideas that the Gaussian conjugate Rogers-Shephard inequality due to Milman-Nakamura-Tsuji yields improved Schmuckenschlaeger-type bounds for intersections and Minkowski sums of centrally symmetric convex bodies.

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